A Multiscale Theory for Image Registration and Nonlinear Inverse Problems

Abstract

In an influential paper, Tadmor, Nezzar and Vese (Multiscale Model. Simul. (2004)) introduced a hierarchical decomposition of an image as a sum of constituents of different scales. Here we construct analogous hierarchical expansions for diffeomorphisms, in the context of image registration, with the sum replaced by composition of maps. We treat this as a special case of a general framework for multiscale decompositions, applicable to a wide range of imaging and nonlinear inverse problems. As a paradigmatic example of the latter, we consider the Calder\'on inverse conductivity problem. We prove that we can simultaneously perform a numerical reconstruction and a multiscale decomposition of the unknown conductivity, driven by the inverse problem itself. We provide novel convergence proofs which work in the general abstract settings, yet are sharp enough to settle an open problem on the hierarchical decompostion of Tadmor, Nezzar and Vese for arbitrary functions in $L^2$. We also give counterexamples that show the optimality of our general results.